Table Of ContentPOLYHEDRAL DIVISORS AND SL2-ACTIONS
ON AFFINE T-VARIETIES
IVANARZHANTSEVAND ALVAROLIENDO
Abstract. InthispaperweclassifySL2-actionsonnormalaffineT-varietiesthatarenormalized
by the torus T. This is done in terms of a combinatorial description of T-varieties given by
AltmannandHausen. Themainingredientisageneralization ofDemazure’srootsofthefanof
2 a toric variety. As an application we give a description of special SL2-actions on normal affine
1 varieties. We also obtain, in our terms, the classification of quasihomogeneous SL2-threefolds
0 dueto Popov.
2
n
a
J
8 Introduction
2
Let k be an algebraically closed field of characteristic zero, M be a lattice of rank n, N =
] Hom(M,Z) be the dual lattice of M, and T be the algebraic torus Speck[M], so that M is the
G
character lattice of T and N is the one-parameter subgroup lattice of T.
A
A T-variety X is a normal algebraic variety endowed with an effective regular action of T.
.
h The complexity of a T-action is the codimension of a general orbit, and since the T-action on X
t is effective, the complexity of X equals dimX −rankM. For an affine variety X, to introduce
a
m a T-action on X is the same as to endow k[X] with an M-grading. There are well known
[ combinatorial descriptions of T-varieties. We send the reader to [Dem70] and [Ful93] for the
case of toric varieties, to [KKMS73, Ch. 2 and 4] and [Tim08] for the complexity one case, and
2
to [AH06, AHS08] for the general case. In this paper we use the approach in [AH06].
v
4 Any affine toric variety is completely determined by a polyhedral cone σ ⊆ N . Similarly,
Q
9 the description of a normal affine T-varieties X due to Altmann and Hausen [AH06] involves
4 the data (Y,σ,D) where Y is a normal semiprojective variety, σ ⊆ N := N ⊗Q is a polyhedral
4 Q
. cone, and D is a divisor on Y whose coefficients are polyhedra in NQ with tail cone σ. The
5
divisor D is called a σ-polyhedral divisor on Y (see Section 1.1 for details).
0
1 Let X be a T-variety endowed with a regular G-action, where G is any linear algebraic group.
1 We say that the G-action on X is compatible if the image of G in Aut(X) is normalized but not
:
v centralized by T. Furthermore, we say that the G-action is of fiber type if the general orbits are
i contained in the T-orbit closures, and of horizontal type otherwise [FZ05, Lie10a].
X
Let now G = G (k) be the additive group of k. It is well known that a G -action on an
r a a a
a affine variety X is equivalent to a locally nilpotent derivation (LND) of k[X]. A description of
compatible G -actions on an affine T-variety, or equivalently of homogeneous LNDs on k[X], is
a
available in the case where X is of complexity at most one [Lie10a] or the G -action is of fiber
a
type [Lie10b] in terms of a generalization of Demazure’s roots of a fan [Dem70] (see Sections 1.3
and 1.4).
A regular SL -action on an affine variety X is uniquely defined by an sl -triple {δ,∂ ,∂ }
2 2 + −
of derivations of the algebra k[X], where ∂ are locally nilpotent, δ = [∂ ,∂ ] is semisimple
± + −
and [δ,∂ ] = ±2∂ (see Proposition 2.1). Assume now that X is an affine T-variety. If the
± ±
Date: January 23, 2013.
2010 Mathematics Subject Classification: 13N15, 14L30, 14M25, 14R20, 14M17.
Key words: SL2-actions, torus actions, affine varieties, locally nilpotent derivations, special actions, quasi-
homogeneous SL2-threefolds.
The first author was partially supported byRFBR grant 09-01-00648-a and theSimons Foundation.
1
2 IVANARZHANTSEVANDALVAROLIENDO
SL -action is compatible, then ∂ are homogeneous with respect to the M-grading on k[X] and
2 ±
the grading given by δ is a downgrading of the M-grading.
The main result of this paper, contained in Section 2, is a classification of compatible SL -
2
actionsonanaffineT-varietyX inthecasewherethisactionisoffibertypeorX isofcomplexity
one (See Theorems 2.12 and 2.18, respectively). Our idea is to classify compatible SL -actions
2
by calculating the commutator of two homogeneous LNDs. The existence of a compatible SL -
2
action on X puts strong restrictions on the combinatorial data (Y,σ,D) and endows D with an
additional structure. It should be noted that if the T-variety X is of complexity one and the
SL -action is of horizontal type, then X is spherical with respect to a bigger reductive group,
2
namely, an extension of SL by a torus. We do not use the theory of spherical varieties in this
2
paper.
The rest of the paper is devoted to two applications of our main result: special SL -actions
2
and SL -actions with an open orbit. A G-action on X is called special (or horospherical) if there
2
exists adenseopen W ⊆ X suchthattheisotropygroupof anypointx ∈ W contains amaximal
unipotent subgroup of G. Special actions play an important role in Invariant Theory.
Any special action of a connected reductive group G on an affine variety X may be recon-
structed from the action of a maximal torus T ⊆ G on the algebra k[X]U of invariants of a
maximal unipotent subgroup U [Pop86, Theorem 5]. This reduces the study of special actions
to torus actions. In Section 3 we illustrate this phenomenon for SL -actions in our terms (see
2
Theorem3.11 and3.12). Inparticular, weshowthatforeveryspecial SL -action onan affineva-
2
rietyX thereisacanonical2-torusactionandtheSL -actioniscompatibleandoffibertypewith
2
respectto this torus. Sincethe reconstruction of the G-variety X from theT-variety Speck[X]U
is an algebraic procedure, it is useful to have a geometric description of X. In Proposition 3.10
we describe a normal affine variety X with a special SL -action as a T2-variety with respect to
2
thecanonical torus T2. It is worthwhileto remark that any G-action on an affine variety may be
contracted to a special one [Pop86, Proposition 8]. It will be interesting to interpret contraction
of SL -actions in terms of polyhedral divisors.
2
As a corollary of our classification of special actions, we prove that if an affine T-variety X of
complexity one admits a compatible special SL -action of horizontal type, then X is toric with
2
respect to a bigger torus and the SL -action is compatible with respect to the big torus as well.
2
Furthermore, using a linearization result due to Berchtold and Hausen [BH03], we show that,
up to conjugation in Aut(X), any special SL -action on an affine toric 3-fold X is compatible
2
with the big torus, and thus it is given by an SL -root (see Definition 2.6).
2
It is natural to generalize Altmann and Hausen’s approach [AH06] to arbitrary reductive
groups. Special actions formthemostaccessibleclass forsuchageneralization. Ourworkinthis
line may be regarded as a first step towards this aim. It must be noted that Timashev [Tim97]
already gave a combinatorial description for G-actions of complexity one in the framework of
Luna-Vust theory.
Finally, our method allows to reprove, in Section 4, Popov’s classification of generically tran-
sitive SL -actions on normal affinethreefolds. Theonly fact that weuseis the existence of a one
2
dimensional torus R commuting with SL . Together with the maximal torus in SL , this allows
2 2
us to consider a quasi-homogeneous threefold as a T2-variety of complexity one, where T2 is a
two dimensional torus. We also obtain, as a direct consequence of our results, the characteriza-
tion of toric quasi-homogeneous SL -threefolds given in [Ga˘ı08] and [BH08] (see Corollaries 4.9
2
and 4.13). Recall that a G-variety is quasi-homogeneous if it has an open G-orbit.
In the entire paper the term variety means a normal integral scheme of finite type over an
algebraically closed field k of characteristic zero. The term point always refer to a closed point.
The authors are grateful to Mikhail Zaidenberg for useful discussions and the referee for
valuable suggestions. This work was done during stays of both authors at the Institut Fourier,
Grenoble. We thank the Institut Fourier for its support and hospitality.
P-DIVISORS AND SL2-ACTIONS ON T-VARIETIES 3
Contents
Introduction 1
1. Preliminaries 3
1.1. Combinatorial description of T-varieties 3
1.2. Locally nilpotent derivations and G -actions 4
a
1.3. Locally nilpotent derivations on affine toric varieties 5
1.4. Locally nilpotent derivations on affine T-varieties 5
2. Compatible SL -actions on normal affine T-varieties 7
2
2.1. SL -actions on affine varieties 7
2
2.2. SL -actions on toric varieties 9
2
2.3. SL -actions of fiber type on T-varieties 10
2
2.4. SL -actions of horizontal type on T-varieties 11
2
3. Special SL -actions 16
2
4. Quasi-homogeneous SL -threefolds 19
2
4.1. SL -threefolds via polyhedral divisors 19
2
4.2. Parameters 21
4.3. Relation between slope and height 22
4.4. Generically transitive SL ×Ts-action 23
2
Appendix: The commutator formulas 23
References 24
1. Preliminaries
Inthissection werecalltheresultsaboutG -actions onaffineT-varieties neededinthispaper.
a
1.1. Combinatorial description of T-varieties. Let M be a lattice of rank n and N =
Hom(M,Z) be its dual lattice. We let M = M ⊗Q, N = N ⊗Q, and we consider the natural
Q Q
duality pairing M ×N → Q, (m,p) 7→ hm,pi = p(m).
Q Q
Let T = Speck[M] be the n-dimensional algebraic torus associated to M and let X = Spec A
be an affine T-variety. The comorphism A → A ⊗ k[M] induces an M-grading on A and,
conversely, every M-grading on A arises in this way. The T-action on X is effective if and only
if the corresponding M-grading is effective.
In [AH06], a combinatorial description of normal affine T-varieties is given. In what follows
we recall the main features of this description. Let σ be a pointed polyhedral cone in N . We
Q
define Pol (N ) to be the set of all σ-polyhedra i.e., the set of all polyhedra in N that can be
σ Q Q
decomposed as the Minkowski sum of a bounded polyhedron and the cone σ.
Recall that σ∨ stands for the cone in M dual to σ. To a σ-polyhedron ∆ ∈ Pol (N ) we
Q σ Q
associate its support function h : σ∨ → Q defined by
∆
h (m) = minhm,∆i = minhm,pi.
∆
p∈∆
Furthermore, if we let {v } be the set of all vertices of ∆, then the support function is given by
i
h (m) = min{v (m)} for all m ∈ σ∨. (1)
∆ i
i
Hence, h is piecewise linear, concave, and positively homogeneous.
∆
Definition 1.1. A normal variety Y is called semiprojective if it is projective over an affine
variety. A σ-polyhedral divisor on Y is a formal sum D = ∆ ·Z, where Z runs over all
Z Z
prime divisors on Y, ∆ ∈ Pol (N ), and ∆ = σ for all but finitely many Z. For m ∈ σ∨ we
Z σ Q Z
P
4 IVANARZHANTSEVANDALVAROLIENDO
can evaluate D in m by letting D(m) be the Q-divisor
D(m)= h (m)·Z,
Z
Z⊆Y
X
whereh isthesupportfunctionof∆ . Aσ-polyhedraldivisorDiscalledproper ifthefollowing
Z Z
hold:
(i) D(m) is semiample and Q-Cartier for all m ∈ σ∨, and
(ii) D(m) is big for all m ∈ rel.int(σ∨).
Hererel.int(σ∨)denotes therelative interiorof theconeσ∨. Furthermore, aQ-Cartier divisor
D on Y is called semiample if there exists r > 0 such that the linear system |rD| is base point
free, and big if there exists a divisor D ∈ |rD|, for some r > 0, such that the complement
0
Y \SuppD is affine.
0
The following theorem gives a combinatorial description of T-varieties analogous to the clas-
sical combinatorial description of toric varieties. In the sequel, χm denotes the character of T
corresponding to the lattice vector m, and σ∨ denotes the semigroup σ∨∩M. Furthermore, for
M
a Q-divisor D on Y, O (D) stands for the sheaf O (⌊D⌋).
Y Y
Theorem 1.2 ([AH06]). To any proper σ-polyhedral divisor D on a semiprojective variety Y
one can associate a normal affine T-variety of dimension rankM +dimY given by X[Y,D] =
SpecA[Y,D], where
A[Y,D] = A χm, and A = H0(Y,O (D(m)) ⊆k(Y).
m m Y
m∈σ∨
MM
Conversely, any normal affine T-variety is isomorphic to X[Y,D] for some semiprojective
variety Y and some proper σ-polyhedral divisor D on Y.
We call Y the base variety and the pair (Y,D) the combinatorial description of X. We also
define the support of a proper σ-polyhedral divisor as SuppD = {Z ⊆ Y | ∆ 6= σ}.
Z
This combinatorial description is not unique, but can be made unique by adding some mini-
mality conditions on the pair (Y,D), see [AH06, Section 8]. Here we only need a particular case
of [AH06, Corollary 8.12].
Corollary 1.3. Let D and D′ be two proper σ-polyhedral divisors on a normal semiprojective
variety Y. If for every prime divisor Z in Y there exists a vector v ∈ N such that
Z
D= D′+ (v +σ)·Z, and hm,v i·Z is principal, ∀m ∈ σ∨ ,
Z Z M
Z Z
X X
then X[Y,D] is equivariantly isomorphic to X[Y,D′].
Most of this paper deals with the case where the base is a curve C isomorphic to A1 or P1.
Any σ-polyhedral divisor on A1 is proper. If D = ∆ · z is a σ-polyhedral divisor on
z∈C z
C = P1, then D is proper if and only if degD := ∆ ( σ. We also need the following
Pz∈C z
result from [AH06, Section 11].
P
Corollary 1.4. Let D be a proper σ-polyhedral divisor on a smooth curve C. Then X[C,D] is
toric if and only if C = A1 and D can be chosen (via Corollary 1.3) supported in at most one
point, or C = P1 and D can be chosen (via Corollary 1.3) supported in at most two points.
1.2. Locally nilpotent derivations and G -actions. Let X = Spec A be an affine variety.
a
A derivation ∂ on A is called locally nilpotent (LND for short) if for every a ∈ A there exists
n∈ Z such that ∂n(a) = 0. We denote by G the additive group of the base field k. Given an
≥0 a
LND ∂ on A, the map φ : G ×A → A, φ (t,f) = exp(t∂)(f) defines a G -action on X, and
∂ a ∂ a
any G -action on X arises in this way [Fre06].
a
LetnowDbeaproperσ-polyhedraldivisoronasemiprojectivevarietyY,andletA = A[Y,D]
be the corresponding M-graded domain. A G -action on X = SpecA is said compatible with
a
P-DIVISORS AND SL2-ACTIONS ON T-VARIETIES 5
the T-action on X if the image of G in Aut(X) is normalized by the torus T. A G -action
a a
is compatible if and only if the corresponding LND ∂ on A is homogeneous i.e., if ∂ sends
homogeneous elements to homogeneous elements. Any homogeneous LND ∂ has a well defined
degree given as deg∂ = deg∂(f)−degf for any homogeneous f ∈ A\ker∂.
A homogeneous LND ∂ on A extends to a derivation on FracA = k(Y)(M) by the Leibniz
rule, where k(Y)(M) is the field of fractions of k(Y)[M]. The LND ∂ is said to be of fiber type
if ∂(k(Y)) = 0 and of horizontal type otherwise. Geometrically speaking, ∂ is of fiber type if
and only if the general orbits of the corresponding G -action on X = Spec A are contained in
a
the orbit closures of the T-action given by the M-grading.
1.3. Locally nilpotent derivations on affine toric varieties. In this section we recall the
classification of homogeneous LNDs on toric varieties given in [Lie10a]. A similar description is
implicit in [Dem70, Section 4.5]. As usual, for a cone σ, we denote by σ(1) the set of all rays
of σ and we identify a ray with its primitive vector.
Definition 1.5. Let σ be a pointed cone in N . We say that e ∈ M is a root of the cone σ if
Q
the following hold:
(i) there exists ρ ∈ σ(1) such that he,ρ i= −1;
e e
(ii) he,ρi ≥ 0, for all ρ ∈σ(1)\{ρ }.
e
The ray ρ is called the distinguished ray of the root e. We denote by R(σ) the set of all roots
e
of σ.
One easily checks that any ray ρ ∈ σ(1) is the distinguished ray for infinitely many roots
e ∈ R(σ). For every root e ∈ R(σ) we define a homogeneous derivation ∂ of degree e of the
e
algebra k[σ∨ ] by the formula
M
∂ (χm)= hm,ρ i·χm+e, for all m ∈ σ∨ .
e e M
The following theorem gives a classification of the homogeneous LNDs on k[σ∨ ].
M
Theorem 1.6. For every root e ∈ R(σ), the homogeneous derivation ∂ on k[σ∨ ] is an LND
e M
of degree e with kernel ker∂ = k[τ ∩M], where τ is the facet of σ∨ dual to the distinguished
e e e
ray ρ . Conversely, if ∂ 6= 0 is a homogeneous LND on k[σ∨ ], then ∂ = λ∂ for some root
e M e
e∈ R(σ), and some λ ∈ k∗.
1.4. Locally nilpotent derivations on affine T-varieties. We give first a classification of
homogeneous LNDs of fiber type on T-varieties of arbitrary complexity given in [Lie10b].
Letting D = ∆ ·Z be a proper σ-polyhedral divisor on a semiprojective variety Y, we
Z Z
let A = A[Y,D]. For every prime divisor Z ⊆ Y, we let {v | i = 1,··· ,r } be the set of all
i,Z Z
P
vertices of ∆ . Letting e be a root of the cone σ, we define the divisor
Z
D(e) = min{v (e)}·Z, and Φ∗ = H0(Y,O (D(e)))\{0}.
i,Z e Y
i
Z
X
Remark that the evaluation divisor D(m) is only defined for m ∈ σ∨ and e∈/ σ∨. The reason for
the above notation is that taking (1) as the definition of support function, we obtain the above
formula for the evaluation divisor, which can be evaluated at any m ∈M .
Q
For every ϕ ∈ Φ∗ we let
e
∂ (fχm) = hm,ρ i·ϕ·fχm+e, for all m ∈ σ∨ , and f ∈ k(Y).
e,ϕ e M
ThefollowingtheoremgivesaclassificationofthehomogeneousLNDsoffibertypeonA[Y,D].
Theorem 1.7. For every root e ∈ R(σ) and ϕ∈ Φ∗, the derivation ∂ is a homogeneous LND
e e,ϕ
of fiber type on A = A[Y,D] of degree e with kernel
ker∂ = A χm,
e,ϕ m
m∈Mτe∩M
6 IVANARZHANTSEVANDALVAROLIENDO
where τ ⊆ σ∨ is the facet dual to the distinguished ray ρ . Conversely, if ∂ 6=0 isa homogeneous
e e
LND of fiber type on A, then ∂ = ∂ for some root e ∈ R(σ) and some ϕ∈ Φ∗.
e,ϕ e
Theclassification of LNDs of horizontal type is moreinvolvedand is only available in the case
of complexity one. Here, we give an improved presentation of the classification given in [Lie10a,
Theorem 3.28].
Sincethe complexity is one thebase variety Y is asmooth curveC. Let D = ∆ ·z be a
z∈C z
proper σ-polyhedral divisor on C, and let X = X[C,D]. If A = A[C,D] admits a homogeneous
LND of horizontal type, then C is isomorphic either to A1 or to P1. In the folloPwing we assume
that C = A1 or C = P1.
Definition 1.8. A colored σ-polyhedral divisor on C is a collection D = {D;v ,∀z ∈ C} if
z
C = A1 and D = {D,z ;v ,∀z ∈ C \z } if C = P1, satisfying the following conditions:
∞ z ∞
(1) D = ∆ ·z is a proper σ-polyhedral divisor on C, z ∈ C, aend v is a vertex of ∆ .
z∈C z ∞ z z
We let C′e= C if C = A1 and C′ = C \{z } if C = P1.
∞
(2) v :P= v is a vertex of degD| , and
deg z∈C′ z C′
(3) v ∈ N with at most one exception.
z
P
We also let z ∈ C′ be such that v ∈ N for all z ∈ C′\{z }. We say that D is a coloring of
0 z 0
D and we call z the marked point, z the point at infinity if C = P1, and v the colored vertex
0 ∞ z
of the polyhedron ∆ . e
z
Remark that the above notion of coloring is independent from the notion of coloring in the
theory of spherical varieties.
Let D be a colored σ-polyhedral divisor on C. Letting ω ⊆ N be the cone generated by
Q
degD|C′ −vdeg we let ω ⊆ (N ⊕Z)Q be the cone generated by (ω,0) and (vz0,1) if C = A1,
and bye(ω,0), (v ,1) and (∆ + v − v + ω,−1) if C = P1. Denote by d the minimal
z0 z∞ deg z0
positive integer such that d·v ∈ N. We call ω the associated cone of the colored σ-polyhedral
e z0
divisor D.
Definition 1.9. A pair (D,e), where D is a ceolored σ-polyhedral divisor on C and e ∈ M, is
said to be coherent if
(1) There exists s ∈ Z suech that e = (ee,s) ∈ M ⊕ Z is a root of the associated cone ω with
distinguished ray ρ= (d·vz0,d). In this case s = −1/d−vz0(e).
(2) v(e) ≥ 1+v (e), for every z ∈ C′\{z } and every vertex v 6= v of the polyhedron ∆ .
z e 0 z ez
(3) d·v(e) ≥ 1+d·v (e), for every vertex v 6= v of the polyhedron ∆ .
ze0 z0 z0
(4) If Y = P1, then d·v(e) ≥ −1−d·v (e), for every vertex v of the polyhedron ∆ .
deg z∞
Let now L = {m ∈ M | v (m) ∈ Z} and ϕm ∈ k(C) be a rational function with
z0
div(ϕm)| +D(m)| = 0, and ϕm ·ϕm′ = ϕm+m′ for all m,m′ ∈ ω∨.
C′ C′ L
The choice of ϕm as above is possible since D(m) is linear for m ∈ ω∨.
The following theorem gives a classification of homogeneous LNDs of horizontal type on
A[C,D]. It corresponds to [Lie10a, Theorem 3.28].
Theorem 1.10. Let X = X[C,D] be a normal affine T-variety of complexity one. Then the
homogeneous LNDs of horizontal type on k[X] = A[C,D] are in bijection with the coherent
pairs (D,e), where D is a coloring of D and e ∈ M. Furthermore, the homogeneous LND ∂
corresponding to (D,e) has degree e and kernel
e e
ker∂ = kϕm.
e
m∈ω∨
ML
Let us give an explicit formula for the homogeneous LND ∂ associated to the coherent pair
(D,e). Withoutlossofgeneralitywemayassumez = 0andz = ∞ifC = P1. ByCorollary1.3
0 ∞
e
P-DIVISORS AND SL2-ACTIONS ON T-VARIETIES 7
we may assume v = ¯0 ∈ N for all z ∈ C′\{z }. Letting k[C′] = k[t], the homogeneous LND of
z 0
horizontal type ∂ corresponding to the coherent pair (D,e) is given by
∂(χm·tr) = d(v (m)+r)·χm+e·tr+s, for all (m,r) ∈ M ⊕Z. (2)
0
e
Furthermore, if we let m = (m,r) ∈ M ⊕Z and χme = χm·tr, then (2) can be written as in the
toric case
e∂(χme)= hm,ρi·χme+ee, for all m ∈ M ⊕Z. (3)
We also need the following two technical lemmas. They follow from Theorem 1.10.
e e e
Lemma1.11 (cf. [Lie10a,Lemma4.5]). LetX = X[C,D] and let∂ and∂ betwo homogeneous
1 2
LNDs of horizontal type on k[X]. Assume that z (∂ ) = z (∂ ). Then ker∂ ∩ker∂ ) k if
∞ 1 ∞ 2 1 2
and only if ω(∂ )∩ω(∂ ) ) {0}. Furthermore, if rankM = 2 this is the case if and only if the
1 2
vertices v (∂ ) and v (∂ ) are adjacent vertices in the polyhedron degD| .
deg 1 deg 1 C′
Lemma 1.12 (cf. [Lie10a, Remark 3.27]). Let X = X[C,D] and let ∂ be a homogeneous LND
of horizontal type on k[X] of degree e. Then
(1) If C = A1 then e∈ ω∨ ⊆ σ∨.
(2) If C = P1 and for every ray ρ∈ σ(1)∩ω(1) we have ρ∩degD = ∅, then e ∈ ω∨ ⊆ σ∨.
2. Compatible SL -actions on normal affine T-varieties
2
In this section we give a classification of compatible SL -actions on T-varieties in two cases:
2
in the case where the T-action is of complexity one; and in arbitrary complexity provided that
the general SL -orbits are contained in the T-orbit closures.
2
2.1. SL -actions on affine varieties. Let SL be the algebraic group of 2 × 2 matrices of
2 2
determinant 1. Every algebraic subgroup of SL of positive dimension is conjugate to one of the
2
following subgroups:
T = t 0 |t ∈ k∗ , U = ǫ λ | ǫ,λ ∈ k,ǫs = 1 ,
0 t−1 (s) 0 ǫ−1,
(cid:8)(cid:0) N(cid:1)= T ∪ (cid:9)0 1 ·T, ann(cid:16)d B =(cid:17)T ·U . o
−1 0 (1)
Here T is a maximal torus, N is the normalizer of a maximal torus, B is a Borel subgroup,
(cid:0) (cid:1)
and U is a cyclic extension of a maximal unipotent subgroup. We also define the following
(s)
maximal unipotent subgroups:
U = U , and U = 0 −1 ·U · 0 1 .
+ (1) − 1 0 (1) −10
As a group SL is generated by the unipotent subgroups U and U isomorphic to G .
2 (cid:0) (cid:1) + (cid:0) −(cid:1) a
Let nowX = SpecA bean affinevariety endowedwith an SL -action. ThetwoU -actions on
2 ±
X are equivalent to two LNDs ∂ on the algebra A, and the T-action on X is equivalent to a Z-
±
grading on A. Furthermore, this Z-grading on A is also uniquely determined by its infinitesimal
generator i.e., by the semisimple derivation δ given by δ(a) = deg(a)·a for every homogeneous
a ∈ A.
The following well known proposition gives a criterion for the existence of an SL -action on
2
an affine variety. In the lack of a reference we provide a short proof, cf. [FZ05, 4.15].
Proposition 2.1. A non-trivial SL -action on an affine variety X = SpecA is equivalent
2
to a (not necessarily effective) Z-grading on A with infinitesimal generator δ and a couple of
homogeneous LNDs (∂ ,∂ ) of degrees deg ∂ = ±2, satisfying [∂ ,∂ ]= δ.
+ − Z ± + −
Furthermore, the Z-grading is effective if and only if SL acts effectively on X. If the Z-
2
grading is not effective, then the kernel of SL → Aut(X) is {±Id} and so PSL acts effectively
2 2
on X.
Proof. Assume first that SL acts non-trivially on X. Let {h,e ,e } be the sl -triple in the Lie
2 + − 2
algebra sl . Since e is tangent to the 1-parameter unipotent subgroup U in SL , it acts on
2 ± ± 2
8 IVANARZHANTSEVANDALVAROLIENDO
A as an LND ∂ . The vector h is tangent to the torus T, thus h acts on A as the infinitesimal
±
generator δ of a Z-grading. Since [h,e ] = ±2e , the LND ∂ is homogeneous of degree ±2,
± ± ±
and the relation [e ,e ] = h implies [∂ ,∂ ]= δ.
+ − + −
Conversely, assume that we have δ,∂ ,∂ as in the proposition. Then s = hδ,∂ ,∂ i is a
+ − + −
Lie subalgebra in Der(A) isomorphic to sl . Furthermore, every element of A is contained in a
2
finite-dimensional s-submodule. Recall that any finite-dimensional sl -module has a canonical
2
structure of SL -module whose tangent representation coincides with the given one. This gives
2
A the structure of a rational SL -module. Since SL is generated as a group by the subgroups
2 2
U and the LNDs ∂ define the action of U via automorphisms, the group SL acts on A via
± ± ± 2
automorphisms. ThisprovesthatAisarationalSL -algebra, or, equivalently, SL acts regularly
2 2
on X. (cid:3)
We restrict now to the case of affine T-varieties. The following definition determines the class
of SL -actions that we study in the sequel.
2
Definition 2.2. An SL -action on a T-variety X is compatible if the image of SL in Aut(X) is
2 2
normalized but not centralized by the torus T.
Assume now that X is a T-variety endowed with a compatible SL -action. Denote by SL the
2 2
imageofSL inAut(X). Thereisahomomorphismψ : T → Aut(SL ). Sinceanyautomorphism
2 2
from Aut(SL ) is inner, we have Aut(SL ) ≃ PSL . Thus the image of T is either trivigal or is
2 2 2
a maximal torus T ⊆ PSL . In the first case T centralizes SL sgo this case is excluded by the
2 2
definition ogf a compatible SL -action. gHence T contains T and so T = T ·S, where S = kerψ is
2
a complementary subtorus that centralizes the SL2-action. Lget U± be unipotent root subgroups
in SL with respect to the torus T. Then the SL -action on X is defined by the infinitesimal
2 2
generator corresponding to a Z-grading on k[X] defined by T (this is a downgrading of the
M-ggrading) and two M-homogeneous LNDs ∂± corresponding to the U±-actions. This gives the
following corollary.
Corollary 2.3. (i) Let X be a normal affine T-variety endowed with a compatible SL -action.
2
In this case, in Proposition 2.1, we may assume that δ is the infinitesimal generator cor-
responding to a downgrading of M and that ∂ are M-homogeneous LNDs. Furthermore,
±
T = T ·S, where T is the maximal torus in SL and S is a complementary subtorus that
2
centralizes the SL -action.
2
(ii) Let X be a normal affine T-variety endowed with an SL -action that is centralized by T.
2
Then we may extend T by T so that the SL -action is compatible with this bigger torus
2
action.
The following is a generalization of a definition in [Lie10b].
Definition 2.4. WesaythatacompatibleSL -action onaT-varietyisoffiber type ifthegeneral
2
orbits are contained in the T-orbit closures and of horizontal type otherwise.
Clearly, a compatible SL -action is of fiber type if and only if both derivations ∂ are of fiber
2 ±
type. The following lemma shows that a compatible SL -action is of horizontal type if and only
2
if both derivations ∂ are of horizontal type.
±
Lemma 2.5. Consider a compatible SL -action on a T-variety X and assume that the LND ∂
2 +
is of fiber type. Then the SL -action is of fiber type.
2
Proof. Set B = T ·U ⊆ SL . Then the B-action on X is of fiber type i.e., the general B-orbits
+ 2
are contained in the orbit closures of the T-action. We consider two cases.
Case 1: The general SL -orbits on X are 2-dimensional. Then for general x ∈ X one has
2
B·x = SL ·x, and the SL -action is of fiber type.
2 2
Case 2: The general SL -orbits on X are 3-dimensional. Consider a general point x ∈ X and
2
the stabilizer T2 ⊆ T of the subvariety B ·x. Since any automorphism of the group B is inner
x
P-DIVISORS AND SL2-ACTIONS ON T-VARIETIES 9
and the torus T normalizes B, we may find a 1-dimensional subtorus S ⊆ T2 which commutes
x x
with the B-action on B ·x. But the image of the homomorphism ψ : T → Aut(SL ) ≃ PSL is
2 2
a maximal torus, and so the subtorus S is in the kernel of ψ. Hence, S commutes with the
x x
SL -action. In particular, S preserves B ·x and SL ·x, and its action on SL ·xgmay be lifted
2 x 2 2
to the action of a maximal torus S ⊆ SL by right multiplication on SL via a finite covering
2 2
S → S . Butitiseasytocheckthatthe(B×S)-action onSL hasanopenorbit,soS permutes
x 2 x
the general B-orbits on SL ·x. Theis provides a contradiction. (cid:3)
2
e e
2.2. SL -actions on toric varieties. In this section we give a complete classification of com-
2
patible SL -actions on affine toric varieties. Since a toric variety has an open T-orbit, every
2
SL -action on a toric variety is of fiber type.
2
Definition 2.6. Let σ ⊆ N be a polyhedral cone. A root e ∈ R(σ) is called an SL -root if
Q 2
also −e∈ R(σ).
If e is an SL -root, then he,ρ i= −1, he,ρ i = 1, and he,ρi = 0 ∀ρ∈ σ(1)\{ρ }. Thus the
2 e −e ±e
numberof SL -roots of a cone σ with r rays does not exceed r(r−1), and this bound is attained
2
for a regular cone of dimension r.
Theorem 2.7. The compatible SL -actions on an affine toric variety X are in bijection with
2 σ
the SL -roots of σ. Furthermore, for every SL -root e ∈ R(σ), the corresponding SL -action is
2 2 2
effective if and only if the lattice vector ρ −ρ is primitive. If ρ −ρ is not primitive, then
−e e −e e
1(ρ −ρ ) is primitive and PSL acts effectively on X .
2 −e e 2 σ
Proof. Let A= k[σ∨ ]and e∈ R(σ) be an SL -root. Letting p = ρ −ρ , wedefinea Z-grading
M 2 −e e
on A via
deg χm = hm,pi ∈ Z, for all m ∈ σ∨ .
Z M
Hence, the infinitesimal generator of the corresponding G -action is given by
m
δ(χm)= hm,piχm, for all m ∈ σ∨ .
M
A routine computation shows that δ, ∂ and ∂ satisfy the conditions of Proposition 2.1.
e −e
Furthermore, since he,pi = 2 then p is primitive or p/2 is primitive. This proves the“only if”
part of the theorem.
Toprovetheconverse,letδ,∂ ,∂ bethreehomogeneousderivationssatisfyingtheconditions
− +
of Proposition 2.1. Since ∂ are LNDs, then ∂ = λ ∂ for some λ ∈ k∗ and some roots
± ± ± e± ±
e ∈ R(σ). Furthermore, since the derivation δ comes from a downgrading of the M-grading
±
on A, there is a lattice element p such that
δ(χm)= hm,piχm, for all m ∈ σ∨ .
M
Since the commutator [∂ ,∂ ] is a homogeneous operator of degree e +e , we have e :=
+ − + −
e = −e . One checks that the commutator is given by
+ −
[∂ ,∂ ](χm) = λ λ hm,ρ −ρ iχm, for all m ∈ σ∨ .
+ − + − −e e M
Hence p =ρ −ρ , λ = λ−1, and the result follows. (cid:3)
−e e + −
Remark 2.8. IfeisanSL -root ofσ, then−eisalsoanSL -root. ThecorrespondingSL -actions
2 2 2
are conjugate.
Example 2.9. Let X be an affine toric variety of dimension 2. Up to automorphism of the
σ
lattice N we may assume that σ ⊆ N is the cone spanned by the vectors ρ = (1,0) and
Q 1
ρ = (a,b), where a ≥ 0, a < b, and gcd(a,b) = 1. By Theorem 2.7 X admits a compatible
2 σ
SL -action if and only if there exists e ∈ M such that he,ρ i = 1 and he,ρ i = −1. The only
2 1 2
solution is e = (1,−1) and b = a+1. Furthermore, the action is effective if and only if b is odd.
It is well known that the toric variety X corresponds to the affine cone over the rational
σ
normal curve C of degree a+1 (also known as Veronese cone). The curve C is the image of P1
10 IVANARZHANTSEVANDALVAROLIENDO
under the morphism
ψ : P1 ֒→ Pa+1, [x :y]7→ xa+1 :xay :xa−1y2 : ... : ya+1 .
TheSL -action onX isinducedbythecanonicalSL -action onthesimpleSL -moduleV(a+1)
2 σ (cid:2) 2 (cid:3) 2
of binary forms of degree a+1.
Example 2.10. Let now X be an affine toric variety of dimension 3. Letting e be an SL -root
σ 2
of σ, we let ρ and ρ be the corresponding distinguished rays and we consider a ray ρ 6= ρ .
e −e ±e
Since he,ρi = 0 it follows that there are at most 2 non-distinguished rays. Thus the cone σ has
at most 4 rays.
Assumefirstthat σ is simplicial and set e= (1,0,0). Then,up to automorphismof thelattice
N, the cone σ is spanned by the vectors ρ = (1,0,0), ρ = (0,1,0), and ρ = (−1,b,a), where
1 2 3
a > 0 and 0 ≤ b < a.
Let now σ be a non simplicial cone and set again e = (1,0,0). Then, up to automorphism of
the lattice N, the cone σ is spanned by the vectors ρ = (1,0,0), ρ = (0,1,0), ρ = (0,b,a),
1 2 3
and ρ = (−1,c,d), where a > 0, 0 ≤ b < a, gcd(a,b) = 1, d > 0, and ac> bd.
4
Remark 2.11. In dimension 4 or greater, a cone admitting an SL -root can have an arbitrary
2
number of rays.
2.3. SL -actions of fiber type on T-varieties. In the following theorem we give a classifica-
2
tion of SL -actions of fiber type on normal affine T-varieties of arbitrary complexity.
2
Theorem 2.12. Let D be a proper σ-polyhedral divisor on a semiprojective variety Y. Then the
compatible SL -actions of fiber type on the affine T-variety X = X[Y,D] are in bijection with
2
the SL -roots e of σ such that the divisor D(e) is principal and D(e)+D(−e) = 0.
2
Furthermore, the corresponding SL -action is effective if and only if the lattice vector ρ −ρ
2 −e e
is primitive. If ρ −ρ is not primitive, then 1(ρ −ρ ) is primitive and PSL acts effectively
−e e 2 −e e 2
on X.
Proof. Letting A = A[Y,D] = k[X], we let e be an SL -root of σ satisfying the conditions of
2
the theorem. As in the toric case, we let ρ ∈ σ(1) be the distinguished rays of the roots ±e,
±e
respectively. Letting p = ρ −ρ , we define a Z-grading on A via
−e e
deg (A ·χm)= hm,pi ∈ Z, for all m ∈ σ∨ .
Z m M
So the infinitesimal generator of the corresponding G -action is given by
m
δ(fχm) = hm,pi·fχm, for all m ∈ σ∨ and f ∈ A .
M m
Letting ϕ be any rational function on Y such that div(ϕ)+D(e) = 0, we let ∂ = ϕ∂ and
+ e
∂ = ϕ−1∂ . The derivations ∂ are LNDs on A by Theorem 1.7.
− −e ±
Now a routinecomputation shows that δ, ∂ and ∂ satisfy the conditions of Proposition 2.1.
+ −
Furthermore, since he,pi = 2 then p is primitive or p/2 is primitive. This proves the“only if”
part of the theorem.
Toprovetheconverse,letδ,∂ ,∂ bethreehomogeneousderivationssatisfyingtheconditions
− +
of Proposition 2.1. Since ∂ are LNDs of fiber type, then ∂ =∂ for some roots e ∈ R(σ)
± ± e±,ϕ± ±
and some ϕ ∈Φ∗ . Similar to the toric case, we can prove that e:= e = −e .
± e± + −
The commutator [∂ ,∂ ] is given by
+ −
[∂ ,∂ ](fχm)= ϕ ϕ hm,pi·fχm, for all m ∈ σ∨ and f ∈A ,
+ − + − M m
where p = ρ −ρ . Hence ϕ = ϕ−1. Furthermore, since ϕ ∈Φ∗ , we have
−e e + − ± e±
div(ϕ )+D(e) ≥ 0, and div(ϕ )+D(−e) ≥0, so that D(e)+D(−e)≥ 0. (4)
+ −
Moreover,
D(e)+D(−e)= min{v (e)}−max{v (e)} ·Z ≤ 0.
i,Z i,Z
i i
Z (cid:18) (cid:19)
X
